Even tiny gaps between your tub and the wall tiles can let enough water in to damage your walls and host mold. Since new caulking will not adhere to old, the only solution is to remove it and do over.
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Latex, acrylic, or PVA caulking is water-based and can be removed with a single-sided razor blade or a utility knife. Use a heat gun or hairdryer on a low setting to help to loosen stubborn sections and pry them out.
Silicone caulking is much harder to remove, which is why pros don't recommend it for tubs.
To apply water-based caulk, make sure the area is squeaky clean and completely dry. Use rubbing alcohol to remove old caulking residue and soap scum so you don't compromise the new seal. Have a bucket of water and a sponge ready for your fingers and for cleaning up any excess.
Cut the tip of the caulking tube at a 45-degree angle wide enough for a 1/4' bead. Hold the tube at a consistent angle in the corner and move at a steady pace along the whole seam, then go back and smooth it with a moistened finger if you need to.
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For more on bathtubs, consider:
Quick Tip: Bathtub Installation
Free-Standing Tubs: Soaking Up the Luxury
Bathroom Essentials: Tubs, Showers, and Sinks